Idempotents in cyclic codes / by Benjamin Brame.
| Author/creator | Brame, Benjamin |
| Other author | Robinson, Zachary. |
| Other author | East Carolina University. Department of Mathematics. |
| Format | Theses and dissertations |
| Publication Info | [Greenville, N.C.] : East Carolina University, 2012. |
| Description | 54 pages : digital, PDF file |
| Supplemental Content | Access via ScholarShip |
| Subjects |
| Summary | Cyclic codes give us the most probable method by which we may detect and correct data transmission errors. These codes depend on the development of advanced mathematical concepts. It is shown that cyclic codes, when viewed as vector subspaces of a vector space of some dimension n over some finite field F, can be approached as polynomials in a ring. This approach is made possible by the assumption that the set of codewords is invariant under cyclic shifts, which are linear transformations. Developing these codes seems to be equivalent to factoring the polynomial x[superscript]n-x over F. Each factor then gives us a cyclic code of some dimension k over F. Constructing factorizations of x[superscript]n-x is accomplished by using cyclotomic polynomials and idempotents of the code algebra. The use of these two concepts together allows us to find cyclic codes in F[superscript]n. Hence, the development of cyclic codes is a journey from codewords and codes to fields and rings and back to codes and codewords. |
| General note | Presented to the faculty of the Department of Mathematics. |
| General note | Advisor: Zachary Robinson. |
| General note | Title from PDF t.p. (viewed July 12, 2012). |
| Dissertation note | M.A. East Carolina University 2012. |
| Bibliography note | Includes bibliographical references. |
| Technical details | System requirements: Adobe Reader. |
| Technical details | Mode of access: World Wide Web. |